At the central fixed point x̄₀ = (c₀/n)(1,…,1)ᵀ, the linearized Jacobian of the hypercycle under constant organization yields a set of normal modes. The eigenvalues of this Jacobian determine the local dynamical character. Crucially, the same equilibrium point changes its stability type as the cycle dimension n increases:

• For , the eigenvalues are real and negative, yielding a focus (a stable node-like attraction).
• For , the eigenvalues become complex with negative real part, producing a spiral sink (trajectories spiral inward).
• For , the eigenvalues are purely imaginary, giving a center (neutral, closed orbits).
• For , the fixed point becomes a saddle with spiral components (some directions attract, others repel).

This classification illustrates how dimensionality alone can alter the qualitative dynamics without moving the equilibrium. The normal modes correspond to eigenvectors of the Jacobian, and their eigenvalues determine whether perturbations grow, decay, or oscillate.

Lyapunov functions prove asymptotic stability of the central fixed point for ,3,4. For , the saddle character prevents such global stability, and numerical integration reveals a stable limit cycle, sustaining cooperative growth.